eliphant, the Muc-off sounds like a good start.

Yes, take the bike in for a check up and while you're there have them show you how to clean the bike and/or get supplies.

My routine, by no means the best is kinda this:
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1) admire bike.

2) use shop cloth or chamois to dust bike, check for scratches and or rust (one of these days I'll fix/paint that on the MTB)

3) take wire brush and lightly brush the chain/cogs/deraileure

4) spray a little WD40 on clean shop cloth and wipe the chain (some here will swear never use WD on a bike, my shop says this gets more gunk off, like pre-rinsing dishes, this is optional)

5) fill Park chain clean tool to level with Citrus degreaser and run the chain backwards through it till squeaky clean.

6) let dry

7) run a line of chain lube on the chain, pedal backwards (bike is on bike rack) and shift through the gears so the lube is distibuted evenly.

8) optional step, clean the frame with frame cleaner or other product

9) admire clean bike.

10) ride till you need to repeat steps 1-9